[1] We present a two-dimensional Glacier Drainage System model (GlaDS) that couples distributed and channelized subglacial water flow. Distributed flow occurs through linked cavities that are represented as a continuous water sheet of variable thickness. Channelized flow occurs through Röthlisberger channels that can form on any of the edges of a prescribed, unstructured network of potential channels. Water storage is accounted for in an englacial aquifer and in moulins, which also act as point sources of water to the subglacial system. Solutions are presented for a synthetic topography designed to mimic an ice sheet margin. For low discharge, all the flow is accommodated in the sheet, whereas for sufficiently high discharge, the model exhibits a channelization instability which leads to the formation of a self-organized channel system. The random orientation of the network edges allows the channel system geometry to be relatively unbiased, in contrast to previous structured grid-based models. Under steady conditions, the model supports the classical view of the subglacial drainage system, with low pressure regions forming around the channels. Under diurnally varying input, water flows in and out of the channels, and a rather complex spatiotemporal pattern of water pressures is predicted. We explore the effects of parameter variations on the channel system topology and mean effective pressure. The model is then applied to a mountain glacier and forced with meltwater calculated by a temperature index model. The results are broadly consistent with our current understanding of the glacier drainage system and demonstrate the applicability of the model to real settings.

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Models are proposed for channelized and distributed flow of meltwater at the base of an ice sheet. The volumes of both channel and distributed systems evolve according to a competition between processes that open drainage space (e.g. sliding over bedrock, melting of the ice) and processes that close it (e.g. viscous creep of the ice due to a positive effective pressure). Channels are generally predicted to have lower water pressure and therefore capture water from the surrounding regions of distributed flow. There is a natural length scale associated with the distributed system that determines the width of the bed from which water can be drawn into a channel. It is suggested that this determines the spacing between major channels and that this may be reflected in the spacing of eskers. A more permeable distributed system results in more widely spaced, and therefore larger, channels. Calculations of the flow into the head of a channel reveal that there is a critical discharge necessary for it to form, and provide a criterion for where channels can exist.

Uncertainty remains about how the surface hydrology of the Greenland ice sheet influences its subglacial drainage system, affecting basal water pressures and ice velocities, particularly over intraseasonal and interseasonal timescales. Here we apply a high spatial (200 m) and temporal (1 h) resolution subglacial hydrological model to a marginal (extending~25 km inland), land-terminating,~200 km 2 domain in the Paakitsoq region, West Greenland. The model is based on that by Hewitt (2013) but adapted for use with both real topographic boundary conditions and calibrated modeled water inputs. The inputs consist of moulin hydrographs, calculated by a surface routing and lake-filling/draining model, which is forced with distributed runoff from a surface energy-balance model. Results suggest that the areal density of lake-bottom moulins and their timing of opening during the melt season strongly affects subglacial drainage system development. A higher moulin density causes an earlier onset of subglacial channelization (i.e., water transport through channels rather than the distributed sheet), which becomes relatively widespread across the bed, whereas a lower moulin density results in a later onset of channelization that becomes less widespread across the bed. In turn, moulin density has a strong control on spatial and temporal variations in subglacial water pressures, which will influence basal sliding rates, and thus ice motion. The density of active surface-to-bed connections should be considered alongside surface melt intensity and extent in future predictions of the ice sheet's dynamics.

We present a continuum model for melt water drainage through a spatially distributed system of connected subglacial cavities, and consider in this context the complications introduced when effective pressure or water pressure drops to zero. Instead of unphysically allowing water pressure to become negative, we model the formation of a partially vapour- or air-filled space between ice and bed. Likewise, instead of allowing sustained negative effective pressures, we allow ice to separate from the bed at zero effective pressure. The resulting model is a free boundary problem in which an elliptic obstacle problem determines hydraulic potential, and therefore also determines regions of zero effective pressure and zero water pressure. This is coupled with a transport problem for stored water, and the coupled system bears some similarities with Hele-Shaw and squeeze-film models. We present a numerical method for computing time-dependent solutions, and find close agreement with semi-analytical travelling wave and steady-state solutions. As may be expected, we find that ice–bed separation is favoured by high fluxes and low ice surface slopes and low bed slopes, while partially filled cavities are favoured by low fluxes and high slopes. At the boundaries of regions with zero water or effective pressure, discontinuities in water level are frequently present, either in the form of propagating shocks or as stationary hydraulic jumps accompanied by discontinuities in potential gradient.

We consider a nonlinear diffusion equation describing the planar spreading of a viscous fluid injected between an elastic sheet and an underlying rigid plane. The dynamics depends sensitively on the physical conditions at the contact line where the sheet is lifted off the plane by the fluid. We explore two possibilities for these conditions (or “regularisations”): a pre-wetted film and a constant-pressure fluid lag (a gas-filled gap between the fluid edge and the contact line). For both flat and inclined planes, we compare numerical and asymptotic solutions, identifying the distinct stages of evolution and the corresponding characteristic rates of spreading.

We present the first general theory of glacier surging that includes both temperate and polythermal glacier surges, based on coupled mass and enthalpy budgets. Enthalpy (in the form of thermal energy and water) is gained at the glacier bed from geothermal heating plus frictional heating (expenditure of potential energy) as a consequence of ice flow. Enthalpy losses occur by conduction and loss of meltwater from the system. Because enthalpy directly impacts flow speeds, mass and enthalpy budgets must simultaneously balance if a glacier is to maintain a steady flow. If not, glaciers undergo out-of-phase mass and enthalpy cycles, manifest as quiescent and surge phases. We illustrate the theory using a lumped element model, which parameterizes key thermodynamic and hydrological processes, including surface-to-bed drainage and distributed and channelized drainage systems. Model output exhibits many of the observed characteristics of polythermal and temperate glacier surges, including the association of surging behaviour with particular combinations of climate (precipitation, temperature), geometry (length, slope) and bed properties (hydraulic conductivity). Enthalpy balance theory explains a broad spectrum of observed surging behaviour in a single framework, and offers an answer to the wider question of why the majority of glaciers do not surge.

We present a new model of subglacial drainage incorporating flow in a network of channels and a porous sheet, with water exchange between the two determined by pressure gradients. The sheet represents the average effect of many linked cavities, whilst the channels emerge from individual cavities that enlarge due to dissipation-induced melting. The model distinguishes cases when the water pressure drops to zero, in which case it allows for the drainage space to be only partially filled with water (free surface flow), and when the pressure reaches the ice overburden pressure, in which case it allows for uplift of the ice to whatever extent is needed to accommodate the water (flotation). Numerical solutions are found for a one-dimensional flow-line version of the model. The results capture typically observed or inferred features of subglacial drainage systems, including open channel flow at the ice margin, seasonal channel evolution, and high water pressures and uplift of the ice surface driven by rapid changes in water supply.